3.8.49 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx\) [749]

Optimal. Leaf size=198 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{5/2}} \]

[Out]

2/9*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)/(e*x+d)^(5/2)/(g*x+f)^(9/2)+8/63*c*d*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)^2/(e*x+d)^(5/2)/(g*x+f)^(7/2)+16/315*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(5/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(5/2)/(g*x+f)^(5/2)

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Rubi [A]
time = 0.15, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {886, 874} \begin {gather*} \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(11/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(9/2)) + (8*c*d
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (16*c
^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(5/2))

Rule 874

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d
*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && EqQ[m - n - 2, 0]

Rule 886

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))),
x] - Dist[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g))), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c
*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*
d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {(4 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{9 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx}{63 (c d f-a e g)^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{9 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{9/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{63 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{315 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 113, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^3 ((a e+c d x) (d+e x))^{3/2} \left (35 g^2-\frac {90 c d g (f+g x)}{a e+c d x}+\frac {63 c^2 d^2 (f+g x)^2}{(a e+c d x)^2}\right )}{315 (c d f-a e g)^3 (d+e x)^{3/2} (f+g x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(11/2)),x]

[Out]

(2*(a*e + c*d*x)^3*((a*e + c*d*x)*(d + e*x))^(3/2)*(35*g^2 - (90*c*d*g*(f + g*x))/(a*e + c*d*x) + (63*c^2*d^2*
(f + g*x)^2)/(a*e + c*d*x)^2))/(315*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(9/2))

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Maple [A]
time = 0.14, size = 172, normalized size = 0.87

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-20 a c d e \,g^{2} x +36 c^{2} d^{2} f g x +35 a^{2} e^{2} g^{2}-90 a c d e f g +63 f^{2} c^{2} d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{315 \left (g x +f \right )^{\frac {9}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}\)	\(169\)
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (8 g^{2} x^{3} c^{3} d^{3}-12 a \,c^{2} d^{2} e \,g^{2} x^{2}+36 c^{3} d^{3} f g \,x^{2}+15 a^{2} c d \,e^{2} g^{2} x -54 a \,c^{2} d^{2} e f g x +63 c^{3} d^{3} f^{2} x +35 a^{3} e^{3} g^{2}-90 a^{2} c d \,e^{2} f g +63 a \,c^{2} d^{2} e \,f^{2}\right ) \left (c d x +a e \right )}{315 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {9}{2}} \left (a e g -c d f \right )^{3}}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(9/2)*(8*c^3*d^3*g^2*x^3-12*a*c^2*d^2*e*g^2*x^2+36*c^
3*d^3*f*g*x^2+15*a^2*c*d*e^2*g^2*x-54*a*c^2*d^2*e*f*g*x+63*c^3*d^3*f^2*x+35*a^3*e^3*g^2-90*a^2*c*d*e^2*f*g+63*
a*c^2*d^2*e*f^2)*(c*d*x+a*e)/(a*e*g-c*d*f)^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((g*x + f)^(11/2)*(x*e + d)^(3/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 968 vs. \(2 (183) = 366\).
time = 1.05, size = 968, normalized size = 4.89 \begin {gather*} \frac {2 \, {\left (8 \, c^{4} d^{4} g^{2} x^{4} + 36 \, c^{4} d^{4} f g x^{3} + 63 \, c^{4} d^{4} f^{2} x^{2} + 35 \, a^{4} g^{2} e^{4} + 10 \, {\left (5 \, a^{3} c d g^{2} x - 9 \, a^{3} c d f g\right )} e^{3} + 3 \, {\left (a^{2} c^{2} d^{2} g^{2} x^{2} - 48 \, a^{2} c^{2} d^{2} f g x + 21 \, a^{2} c^{2} d^{2} f^{2}\right )} e^{2} - 2 \, {\left (2 \, a c^{3} d^{3} g^{2} x^{3} + 9 \, a c^{3} d^{3} f g x^{2} - 63 \, a c^{3} d^{3} f^{2} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{315 \, {\left (c^{3} d^{4} f^{3} g^{5} x^{5} + 5 \, c^{3} d^{4} f^{4} g^{4} x^{4} + 10 \, c^{3} d^{4} f^{5} g^{3} x^{3} + 10 \, c^{3} d^{4} f^{6} g^{2} x^{2} + 5 \, c^{3} d^{4} f^{7} g x + c^{3} d^{4} f^{8} - {\left (a^{3} g^{8} x^{6} + 5 \, a^{3} f g^{7} x^{5} + 10 \, a^{3} f^{2} g^{6} x^{4} + 10 \, a^{3} f^{3} g^{5} x^{3} + 5 \, a^{3} f^{4} g^{4} x^{2} + a^{3} f^{5} g^{3} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{7} x^{6} - a^{3} d f^{5} g^{3} + {\left (15 \, a^{2} c d f^{2} g^{6} - a^{3} d g^{8}\right )} x^{5} + 5 \, {\left (6 \, a^{2} c d f^{3} g^{5} - a^{3} d f g^{7}\right )} x^{4} + 10 \, {\left (3 \, a^{2} c d f^{4} g^{4} - a^{3} d f^{2} g^{6}\right )} x^{3} + 5 \, {\left (3 \, a^{2} c d f^{5} g^{3} - 2 \, a^{3} d f^{3} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c d f^{6} g^{2} - 5 \, a^{3} d f^{4} g^{4}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{6} x^{6} - a^{2} c d^{2} f^{6} g^{2} + {\left (5 \, a c^{2} d^{2} f^{3} g^{5} - a^{2} c d^{2} f g^{7}\right )} x^{5} + 5 \, {\left (2 \, a c^{2} d^{2} f^{4} g^{4} - a^{2} c d^{2} f^{2} g^{6}\right )} x^{4} + 10 \, {\left (a c^{2} d^{2} f^{5} g^{3} - a^{2} c d^{2} f^{3} g^{5}\right )} x^{3} + 5 \, {\left (a c^{2} d^{2} f^{6} g^{2} - 2 \, a^{2} c d^{2} f^{4} g^{4}\right )} x^{2} + {\left (a c^{2} d^{2} f^{7} g - 5 \, a^{2} c d^{2} f^{5} g^{3}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{5} x^{6} - 3 \, a c^{2} d^{3} f^{7} g + {\left (5 \, c^{3} d^{3} f^{4} g^{4} - 3 \, a c^{2} d^{3} f^{2} g^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{3} f^{5} g^{3} - 3 \, a c^{2} d^{3} f^{3} g^{5}\right )} x^{4} + 10 \, {\left (c^{3} d^{3} f^{6} g^{2} - 3 \, a c^{2} d^{3} f^{4} g^{4}\right )} x^{3} + 5 \, {\left (c^{3} d^{3} f^{7} g - 6 \, a c^{2} d^{3} f^{5} g^{3}\right )} x^{2} + {\left (c^{3} d^{3} f^{8} - 15 \, a c^{2} d^{3} f^{6} g^{2}\right )} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(11/2),x, algorithm="fricas")

[Out]

2/315*(8*c^4*d^4*g^2*x^4 + 36*c^4*d^4*f*g*x^3 + 63*c^4*d^4*f^2*x^2 + 35*a^4*g^2*e^4 + 10*(5*a^3*c*d*g^2*x - 9*
a^3*c*d*f*g)*e^3 + 3*(a^2*c^2*d^2*g^2*x^2 - 48*a^2*c^2*d^2*f*g*x + 21*a^2*c^2*d^2*f^2)*e^2 - 2*(2*a*c^3*d^3*g^
2*x^3 + 9*a*c^3*d^3*f*g*x^2 - 63*a*c^3*d^3*f^2*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)
*sqrt(x*e + d)/(c^3*d^4*f^3*g^5*x^5 + 5*c^3*d^4*f^4*g^4*x^4 + 10*c^3*d^4*f^5*g^3*x^3 + 10*c^3*d^4*f^6*g^2*x^2
+ 5*c^3*d^4*f^7*g*x + c^3*d^4*f^8 - (a^3*g^8*x^6 + 5*a^3*f*g^7*x^5 + 10*a^3*f^2*g^6*x^4 + 10*a^3*f^3*g^5*x^3 +
5*a^3*f^4*g^4*x^2 + a^3*f^5*g^3*x)*e^4 + (3*a^2*c*d*f*g^7*x^6 - a^3*d*f^5*g^3 + (15*a^2*c*d*f^2*g^6 - a^3*d*g
^8)*x^5 + 5*(6*a^2*c*d*f^3*g^5 - a^3*d*f*g^7)*x^4 + 10*(3*a^2*c*d*f^4*g^4 - a^3*d*f^2*g^6)*x^3 + 5*(3*a^2*c*d*
f^5*g^3 - 2*a^3*d*f^3*g^5)*x^2 + (3*a^2*c*d*f^6*g^2 - 5*a^3*d*f^4*g^4)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^6*x^6 - a^2
*c*d^2*f^6*g^2 + (5*a*c^2*d^2*f^3*g^5 - a^2*c*d^2*f*g^7)*x^5 + 5*(2*a*c^2*d^2*f^4*g^4 - a^2*c*d^2*f^2*g^6)*x^4
+ 10*(a*c^2*d^2*f^5*g^3 - a^2*c*d^2*f^3*g^5)*x^3 + 5*(a*c^2*d^2*f^6*g^2 - 2*a^2*c*d^2*f^4*g^4)*x^2 + (a*c^2*d
^2*f^7*g - 5*a^2*c*d^2*f^5*g^3)*x)*e^2 + (c^3*d^3*f^3*g^5*x^6 - 3*a*c^2*d^3*f^7*g + (5*c^3*d^3*f^4*g^4 - 3*a*c
^2*d^3*f^2*g^6)*x^5 + 5*(2*c^3*d^3*f^5*g^3 - 3*a*c^2*d^3*f^3*g^5)*x^4 + 10*(c^3*d^3*f^6*g^2 - 3*a*c^2*d^3*f^4*
g^4)*x^3 + 5*(c^3*d^3*f^7*g - 6*a*c^2*d^3*f^5*g^3)*x^2 + (c^3*d^3*f^8 - 15*a*c^2*d^3*f^6*g^2)*x)*e)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(11/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(11/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 4.48, size = 377, normalized size = 1.90 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {70\,a^4\,e^4\,g^2-180\,a^3\,c\,d\,e^3\,f\,g+126\,a^2\,c^2\,d^2\,e^2\,f^2}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x^2\,\left (6\,a^2\,c^2\,d^2\,e^2\,g^2-36\,a\,c^3\,d^3\,e\,f\,g+126\,c^4\,d^4\,f^2\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^4\,d^4\,x^4}{315\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^3\,d^3\,x^3\,\left (a\,e\,g-9\,c\,d\,f\right )}{315\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {4\,a\,c\,d\,e\,x\,\left (25\,a^2\,e^2\,g^2-72\,a\,c\,d\,e\,f\,g+63\,c^2\,d^2\,f^2\right )}{315\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {4\,f\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {4\,f^3\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {6\,f^2\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/((f + g*x)^(11/2)*(d + e*x)^(3/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((70*a^4*e^4*g^2 + 126*a^2*c^2*d^2*e^2*f^2 - 180*a^3*c*d*e^3*f
*g)/(315*g^4*(a*e*g - c*d*f)^3) + (x^2*(126*c^4*d^4*f^2 + 6*a^2*c^2*d^2*e^2*g^2 - 36*a*c^3*d^3*e*f*g))/(315*g^
4*(a*e*g - c*d*f)^3) + (16*c^4*d^4*x^4)/(315*g^2*(a*e*g - c*d*f)^3) - (8*c^3*d^3*x^3*(a*e*g - 9*c*d*f))/(315*g
^3*(a*e*g - c*d*f)^3) + (4*a*c*d*e*x*(25*a^2*e^2*g^2 + 63*c^2*d^2*f^2 - 72*a*c*d*e*f*g))/(315*g^4*(a*e*g - c*d
*f)^3)))/(x^4*(f + g*x)^(1/2)*(d + e*x)^(1/2) + (f^4*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^4 + (4*f*x^3*(f + g*x)
^(1/2)*(d + e*x)^(1/2))/g + (4*f^3*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (6*f^2*x^2*(f + g*x)^(1/2)*(d + e*
x)^(1/2))/g^2)

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